Final answer:
The solution to the differential equation y'' - y' - 6y = 0 is y = Ae^3x + Be^-2x, where A and B are constants.
Step-by-step explanation:
The differential equation given is y'' - y' - 6y = 0. To find the solution to this equation, we can assume a solution of the form y = e^rx. Plugging this into the equation, we get r^2e^rx - re^rx - 6e^rx = 0. Factoring out e^rx, we get the quadratic equation r^2 - r - 6 = 0. Solving this quadratic equation gives us two roots, r = 3 and r = -2. Therefore, the general solution to the differential equation is y = Ae^3x + Be^-2x, where A and B are constants.